Question

An investment firm, SHB-BC Fund, would like to construct a portfolio. The risk-free rate equals 1.5% and a covariance matrix is, [ 0.001 −0.07 0.02 −0.07 0.03 0.09 0.02 0.09 0.002 ] Answer each question for the SHB-BC fund using the following data: Stocks Weights Returns

GM 0.3 0.7%

IBM 0.3 2.0%

MCD 0.4 1.4%

1) Compute a portfolio return using matrix multiplication.

2) Compute a portfolio standard deviation using matrix multiplication.

3) Find analytically an optimal set of weights to minimize portfolio risk subject to the constraints: ∑ ?? = 1 3 ?=1 and ?? > 0.

Answer 1

1) Computation of Portfolio Return

Portfolio Return can be computed as

where R = Expected Return

x = Proportion of funds invested in each security

j = Return of each security

n = No. of securities

Hence Portfolio Returns = 1.37%

2) Computation of Portfolio Standard Deviation

Portfolio Standard Deviation can be computed as

where = Portfolio Standard Deviation

x = Proportion of funds invested in security X

y = Proportion of funds invested in security Y

j = Covariance between X and Y

n = No. of securities

Hence, the portfolio standard deviation is 1.691%

3) Optimal Portfolio using Sharpe Ratio

Sharpe Ratio can be given by

where Rp is the Return of the Security

Ri is the Risk-free rate of return and

is the standard deviation in case the entire portfolio is made of only the concerned security

Sharpe Ratio of GM = -800

Sharpe Ratio of IBM = 16.6667

Sharpe Ratio of MCD = -50

Hence, First preference should be an investment in IBM followed by GM and MCD. This will maximize the return. However, this will not reduce the portfolio risk because the risk-free rate of return is greater than the expected return from GM and MCD. Hence the fund will be better off by investing in risk-free securities.

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0.3 0.7 2.0 1.4 0.3 0.4 1.37

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0.3 0.3 0.3 0.3 0.001-0.07 0.020.3-0 0.01691 0.07 0.03 0.09 0.3 0.02 0.09 0.0020.3 0.4 0.4 0.4 0.4

Rp - Ri

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